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This article gives details of our proposal to replace ordinary chiral$SU(3)_L\times SU(3)_R$ perturbation theory $\chi$PT$_3$ by 3-flavorchiral-scale perturbation theory $\chi$PT$_\sigma$. In $\chi$PT$_\sigma$,amplitudes are expanded at low energies and small $u,d,s$ quark masses about aninfrared fixed point $\alpha^{}_\mathrm{IR}$ of 3-flavor QCD. At$\alpha^{}_\mathrm{IR}$, the quark condensate $\langle\bar{q}q\rangle_{\mathrm{vac}} \not= 0$ induces nine Nambu-Goldstone bosons:$\pi, K, \eta$ and a $0^{++}$ QCD dilaton $\sigma$. Physically, $\sigma$appears as the $f_{0}(500)$ resonance, a pole at a complex mass with real part$\lesssim m_K$. The $\Delta I=1/2$ rule for nonleptonic $K$-decays is then aconsequence of $\chi$PT$_\sigma$, with a $K_S\sigma$ coupling fixed by data for$\gamma\gamma\rightarrow\pi\pi$ and $K_{S} \to \gamma\gamma$. We estimate$R_\mathrm{IR} \approx 5$ for the nonperturbative Drell-Yan ratio $R =\sigma(e^{+}e^{-}\rightarrow\mathrm{hadrons})/  \sigma(e^{+}e^{-}\rightarrow\mu^{+}\mu^{-})$ at $\alpha^{}_\mathrm{IR}$, andshow that, in the many-color limit, $\sigma/f_0$ becomes a narrow $q\bar{q}$state with planar-gluon corrections. Rules for the order of terms in$\chi$PT$_\sigma$ loop expansions are derived in Appendix A, and extended inAppendix B to include inverse-power Li-Pagels singularities due to externaloperators. This relates to an observation that, for $\gamma\gamma$ channels,partial conservation of the dilatation current is not equivalent to$\sigma$-pole dominance.

$\Delta I=1/2$ rule for kaon decays derived from QCD infrared fixed point